Existence of semilinear relaxation shocks

We establish existence with sharp rates of decay and distance from the Chapman--Enskog approximation of small-amplitude shock profiles of a class of semilinear relaxation systems including discrete velocity models obtained from Boltzmann and other kinetic equations. Our method of analysis is based on the macro--micro decomposition introduced by Liu and Yu for the study of Boltzmann profiles, but applied to the stationary rather than the time-evolutionary equations. This yields a simple proof by contraction mapping in weighted $H^s$ spaces